In this talk we will discuss a new window into the solution of Heun differential equations arising in black hole perturbation theory using the tools of two-dimensional conformal field theory and gauge theories. Kerr Compton amplitudes for massless perturbation of generic spin-weight s, are written in compact form in terms of the so-called Nekrasov-Shatashvili functions; their symmetry properties are also discussed. These are then used as building blocks to study the scattering of two Kerr black holes with generic spin orientation. Comparison to conservative observables for bounded systems computed via first-order gravitational self-force methods are shown.